Is there a higher dimensional analogue to translation and rotation. Translation occurs along an axis, and rotations occurs along a plane. Is there some isometry that occurs along a 3-plane (hyperplane?) and does it have a name? Do these isometries generalize well? That is, are there $\binom{N}{n}$ different $n$-dimension isometries in $N$ space where a rotation would be a 2-dimension isometry?

Are you assuming we're working with the Euclidean metric then?
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Robert MastragostinoOct 23 '12 at 0:05

How does this make a difference? I'm asking purely out of curiosity because I am really intrigued that this makes a difference to the answer!
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Steven-OwenOct 23 '12 at 0:08

I'm not sure if it makes a difference or not; I haven't looked into this much. It seems to generalize this way, according to Wikipedia, which makes sense. Any rotation can be represented by an orthogonal matrix, and you can construct $4\times 4$ orthogonal matrices that have two independent eigenvectors with real eigvenvalues.
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Robert MastragostinoOct 23 '12 at 0:28

Check out Quaternions. Quaternion multiplication is rotation in $4$ dimensions similar to how complex multiplication is rotation in $2$ dimensions. This can be generalized to $n$ dimensions using matrices
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NavinOct 23 '12 at 0:30

1 Answer
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Rotations can occur in $n$-dimensions in the same way. For example, you can have rotation on a sphere, for the case $n = 3$.

In general, these $n$-dimensional rotations are characterized by elements of the special orthogonal group, denoted $SO(n)$, which consist of the set of $n \times n$ matrices $Q$ such that $Q^T Q = I$ and $\det(Q) = 1$. For any point $x \in \mathbb{R}^n$, acting such a rotation on $x$ is given by considering the matrix-vector product $Qx$.

If you want such a rotation $Q$ to have as part of its behavior rotation when restricted to some $2$-dimensional plane, then you just want a $Q = Q_1 \oplus Q_2$, where $Q_1$ is a $2 \times 2$ rotation matrix.